Depends. There is no law on the books which requires a presidential candidate to present their qualification to any state, federal, or local agency. As the person taking issue with his qualifications, if this were a court of law then I believe that as plaintiff the burden of proof would lie with you.
What should have happened, IMHO, would have been for a secretary of state in some state to announce that any candidate who failed to supply proof of eligibility would not be put on the ballot. Even if Barack felt this requirement was aimed unfairly at him, it would be hard to argue that the requirement was unreasonable, nor that a secretary of state should allow someone on the ballot who was, in fact, not eligible. The burden of proof would be on Barack to show eligibility.
As it is, things get much murkier. If Barack takes office and signs a piece of legislation, while widespread doubts about his eligibility remained, anyone prosecuted under such legislation should have standing to raise the issue. Were courts to find otherwise, Civil War II would likely follow.